Construction Of Implicit Representation-Based R-functions And Numerical Integration And Application Over The Corresponding Domains

In the field of computer-aided engineering,isogeometric analysis(IGA)avoids the time-consuming meshing process in traditional finite element method(FEM);however,when the domain becomes complicated,the volume parameterization for analysis has not been well solved.The WEB method does not need to do meshing,thus it can be used for problems in implicitly defined complex domains.Considering the advantages of the implicit representation,this thesis mainly discusses the complex solid modeling based on implicit representation and the numerical integration and application over the implicit domains.R-function is widely used in Boolean operations when constructing complex shapes;however,the classical tool,R0 function has square root.Considering that the representation in spline would be more consistent with the CAD system,Chapter 3 con-siders the representation of R-functions with two implicitly defined primitives in spline form.The main idea is mapping the function ranges of the two primitives into a new coordinate system,constructing the function representation on the new coordinate sys-tem,and getting its coefficients by solving the rules obtained from the corresponding requirements of R-function.By using the B-net method and the constraints required by the R-function,the construction process for the spline R-functions represented in Bezier form is introduced.After that,the construction process for the spline R-function represented in B-spline form,and the equivalence between the two forms together with the relations between their coefficients are also given.In order to achieve efficient calculation for more numbers of primitives when tak-ing Boolean operations,in Chapter 4,implicit primitives with local support are consid-ered as input.The efficient local CSG is considered from two aspects.On one hand,by applying the properties of B-splines,the computation can be sped up by local calcula-tion after the dimension reduction process.On the other hand,for input with a CSG tree structure,local evaluation of the primitives can be achieved by updating the bounding boxes of all nodes in the preprocess.Numerical examples show the efficiency of this method.Chapter 5 proposes a numerical method for the integration over the implicitly de-fined domains.The main concerns are the complex topology and the accuracy of the numerical integration.Interval arithmetic is adopted in the hierarchical framework to identify the boundaries.In this way,the topology of the implicit domain can be cor-rectly maintained.A geometry-based local error estimate is proposed to guide the hier-archical subdivision to get accurate results and save the computational cost.Numerical experiments are taken over different types of implicit domains to demonstrate the ef-fectiveness of this method over the methods without using interval arithmetic and the methods taking subdivisions to a specified level.Finally,Chapter 6 applies different types of spline R-functions to partial differential equations for exhibiting their performance.Spline R-function plays the role of weight functions in the WEB method in solving the Poisson equation with a Dirichlet boundary condition.Among the numerical tests,the spline R-function for two primitives is firstly compared with the classical Ro function in solving the partial differential equations.The numerical results demonstrate the robustness and the feasibility of the spline R-functions as the weight functions in the WEB method.Three dimensional spacial spline R-function and the spline R-function composed by primitives with local supports are also tested.Both of them achieve good numerical results in the application of the WEB method in solving equations.